In Blackadars book in 17.4.2 it says that for each element $x \in KK(A,B)$ there is a Kasparov module $(E,\pi ,T)$ such that $T=T^*$. Now, the argument for that is that if $(E,\pi, T)$ is any Kasparov-module, then $(E, \pi, (T+T^*)/2)$ is a compact perturbation which would prove the claim. However, I don't see why $(E, \pi, (T+T^*)/2)$ is Kasparov-Module, to be precise I don't see why $((T+T^*)/2)^2 -1$ is compact. It's probably really easy, but I just don't see it at the moment.

Thank you